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SCHWARTZ CENTER FOR
ECONOMIC POLICY ANALYSIS
THE NEW SCHOOL
WORKING PAPER 2009-3
Global Warming and Economic Externalities
Armon Rezai, Duncan K. Foley
and Lance Taylor
Schwartz Center for Economic Policy Analysis
Department of Economics
The New School for Social Research
6 East 16th Street, New York, NY 10003
www.economicpolicyresearch.org
FEBRUARY
2009
Suggested Citation: Rezai, Armon, Foley Duncan K. and Taylor, Lance.
(2009) “Global Warming and Economic Externalities.” Schwartz Center for
Economic Policy Analysis and Department of Economics, The New School
for Social Research, Working Paper Series.
Global Warming and Economic Externalities
Armon Rezai, Duncan K. Foley, and Lance Taylor
Schwartz Center for Economic Policy Analysis and Department of Economics
New School for Social Research, 6 East 16th Street, New York, NY 10003
[email protected]
Given the scientific evidence that human emissions of greenhouse gases (GHG)
contribute to global warming which will have real economic consequences through climate
change, and the fact that until recently there is neither a market price for GHG emissions nor
alternate institutions to impose limits on emissions, we regard GHG emissions as an uncorrected
negative externality. Economic equilibrium paths in the presence of such an uncorrected
externality are inefficient; as a consequence there is no real economic opportunity cost to
correcting this externality by mitigating global warming. Mitigation investment using resources
diverted from conventional investments can raise the economic well-being of both current and
future generations. The economic literature on GHG emissions misleadingly focuses attention on
the intergenerational equity aspects of mitigation by using a hybrid constrained optimal path as
the business-as-usual benchmark. We calibrate a simple Keynes-Ramsey growth model to
illustrate the significant potential Pareto-improvement from mitigation investment, and to
explain the equilibrium concept appropriate to modeling an uncorrect negative externality.
Keywords: Global warming; Growth with negative externalities; Optimal economic growth
JEL Classification System: D62, O41, Q54
1 Introduction
Much discussion of the economics of global warming emphasizes the issue of trade-offs
in well-being between present and future generations (Nordhaus 2008; Nordhaus and Boyer,
2000; Stern, 2007). Specifically, is it socially beneficial for present and near future generations
to sacrifice their own consumption to mitigate global warming for the benefit of generations yet
to come?
In this paper we argue that this question is beside the point. Global warming is a negative
externality. Standard welfare analysis shows that all generations can benefit from its mitigation.
Current generations can direct less of their foregone consumption to physical capital formation
and more toward mitigation, thereby maintaining their own levels of welfare while bequeathing a
better mix of conventional capital and stock of greenhouse gases (GHG) in the atmosphere to the
future.
We illustrate this point by solving a business as usual economic growth model calibrated
to current data for the intertemporal allocation of capital by a representative agent with an
uncorrected externality and comparing the results to a solution in which the externality is
corrected. The results show that the correction can represent a Pareto improvement from an
inefficient to an efficient growth path with higher consumption levels and lower environmental
damage.
In much of the literature this simple observation is obscured because the optimal path is
compared to a reference path along which the externality is partially corrected. This reference
path maximizes the present discounted value of the felicity of per-capita consumption subject to
the constraint that mitigation expenditure is equal to zero. This constrained optimum implicitly
includes the marginal social cost of emissions in the representative agent's production and
investment decisions, thus partially internalizing the externality. Comparing this solution to the
true optimum incorrectly directs attention toward inter-generation trade-offs because typically
the constrained optimum shows higher per capita consumption for several decades.
When, on the other hand, the optimal and business as usual paths are compared directly,
the first-order effect of optimal mitigation is a potential increase in per capita consumption in
every time period. Intergenerational equity enters into the problem only as a second-order effect
as the optimal program distributes the potential gains from correcting the externality across
generations in accordance with the representative agent's preference for consumption smoothing.
The comparison of the optimal and constrained optimal paths thus leads to an upward biased
estimate of the economic costs of mitigating global warming.
2 The Global Warming Problem
Human (industrial) production entails emissions of GHG. Given scientific evidence like
the results presented in the 4th report of the Intergovernmental Panel on Climate Change (IPCC),
such emissions impact the world climate negatively. An increase in the concentration of GHG is
projected to increase the mean atmospheric temperature implying a higher frequency of disasters
and natural catastrophes (such as droughts, floods, and heat waves), higher mortality rates, and a
significant loss of biodiversity. These consequences have economic costs, the most apparent
being a loss in the productive capacity of the world economy.
The world climate represents a public good as its benefits (or its consumption) are nonrival and non-excludable. Consequently, global warming is a public bad. The presence of public
goods leads to inefficiencies since economic agents do not perceive the true cost of their actions
and do not equalize (social) marginal costs and benefits. In our case the representative agent is
over-emitting GHG, since she perceives the marginal cost of doing so (to her as an individual) is
zero. Under the perfect foresight assumption, she is able to correctly predict the path of GHG
(mainly CO 2 ) concentrations given her (and everybody else's) consumption, production, and
investment choices. Although she is aware of the collective consequences of her actions, she
thinks her individual contribution to the overall result is negligible. Consequently, she will not
reduce her production-related emissions, either through producing less or investing in mitigation,
because she knows that nobody else will do so (as they believe their actions to be insignificant,
too). All agents end up choosing the same inefficient allocation. This point was made in Foley
(2008). Such socially sub-optimal outcomes are well known from simple strategic games, the
most prominent being the Prisoners' Dilemma.
Given the inefficiency of over-accumulation of GHG stock in the atmosphere as a result
of capital stock accumulation, the world economy is not operating at the intertemporal
production possibility frontier (PPF). Future generations would appreciate lower stocks of CO 2
which implies that current generations should accumulate less conventional capital and consume
more (of it) today. There is no intergenerational trade-off despite the fact that such a trade-off is
posited in most of the global warming related economic publications. The mutual gains can be
illustrated by moving the economy from a point inside the PPF to its boundary. This movement
to an efficient equilibrium can be achieved by cost transparency (which amounts to increasing
the cost of emitting to its true value).
Creating the correct price signal for GHG emissions (by whatever means, including capand-trade permits, Pigouvian taxes, or direct regulation) is sufficient to internalize the negative
externality of global warming. As a result our agent will start to invest into mitigation. These
mitigation costs, however, are small compared with the gain of obviating GHG emissions. As is
shown below, averting climate change can represent a non-trivial Pareto improvement.
3 The Model
The model used here for analyzing the economic aspects of global warming is a standard
Ramsey-Cass-Koopmans model of the economy extended to included GHG. In order to
maximize the comparability of our results with models in the literature based on neoclassical
growth theory, the economy only produces one good using a Cobb-Douglas production function,
F [ K , L] , in conventional capital and effective labor. Effective labor consists of the exogenously
given growth paths of population, N , and Harrod-neutral technical change, B , (which can be
translated into Hicks-neutral technical progress given the Cobb-Douglas technology), according
to L = BN . The state equation for conventional capital, K , is in its standard form with capital
increasing due to investment, I , and decreasing due to (exponential) depreciation at rate δ .
Following Nordhaus and Boyer (2000), the stock of GHG in the atmosphere, CD (for
carbon dioxide, CO 2 , measured in parts per million volume, ppmv), enters the model as an
additional state variable. Its dynamics depend on usable output, Y , and are governed by
production-related emissions, G[Y ] , mitigation efforts, M [m]Y , and (exponential) depreciation
at rate ε . m is the share of usable output invested in mitigation. Mitigation efforts, M [m]Y , are
linear in usable output similar to the abatement cost function, Λ[.] , used in Nordhaus and Boyer
(2000) and Nordhaus (2008).
In the interests of keeping the model as parsimonious as possible, temperature dynamics
are omitted and CD in excess of pre-industrial levels lowers productive capacity directly via
what we term a damage function, Z [CD] . There are no sinks and no time lags. Emissions fully
affect output immediately and directly, Y = Z [CD]F [ K , L] . Mitigation can take the form of
removing existing CD from the atmosphere or by preventing current emissions. Mitigation does
not alter carbon emissions intensity permanently.
Formally, the representative agent allocates shares of output to consumption, c , and
investment, s , of which a certain output share, m , is invested into mitigation, in order to
maximize utility, measured as the discounted present value of the felicity of per capita
consumption over time. Let consumption C[t ] = (1 − s[t ])Y [t − 1] and world output
Y [t ] = Z [CD[t ]]F [ K [t ], L[t ]] , then total utility is
⎡ C[t ] ⎤
U⎢
⎥
t =1 (1 + ρ )
⎣ N [t ] ⎦
These choices are subject to initial values K [0] and CD[0] and the following state equations:
T
U [C[t ], t ] = ∑
1
( t −1)
K [t + 1] = (1 − δ ) K [t ] + ( s[t + 1] − m[t + 1])Y [t ]
CD[t + 1] = (1 − ε )CD[t ] + G[Y [t ]] − M [m[t + 1]]Y [t ]
With λ[t ] as the shadow price of capital and μ[t ] the shadow price of CO 2 in the
C[t ]
atmosphere, both expressed in terms of undiscounted felicity in the period t, and c[t ] =
as
N [t ]
per capita consumption in period t, the adjoined Lagrangian for the above problem is
T
Λ[ K , CD, λ , μ , t ] = ∑
1
(1 + ρ ) ( t −1)
(U [c[t ]] + λ[t ]( K [t ] − (1 − δ ) K [t − 1] + ( s[t ] − m[t ])Y [t − 1]) +
μ[t ](CD[t ] − (1 − ε )CD[t − 1] + G[Y [t − 1]] − M [m[t − 1]]Y [t ]))
t =1
(1)
Note that μ[t ] < 0 , as CD affects production negatively. The (negative of the) current
μ[t ]
dollar price of carbon emissions is given by χ [t ] =
.
λ[t ]
3.1 The optimal case (OPT)
For optimality the following first-order conditions, which simultaneously represent a
social competitive equilibrium, have to hold:
∂ λ Λ = 0 ⇔ K [t ] = (1 − δ ) K [t − 1] + ( s[t ] − m[t ])Y [t − 1]
(2)
∂ μ Λ = 0 ⇔ CD[t ] = (1 − ε )CD[t − 1] +
G[Y [t − 1]] − M [m[t ]]Y [t − 1]
(3)
∂ K Λ = 0 ⇔ λ[t ] =
λ[t + 1]
1+ ρ
(1 − δ + rK (1 − m[t + 1] + (G′[Y [t ]] − M [m[t + 1]]) χ [t + 1]) )
λ[t + 1]
(rCD (1 − m[t + 1]) +
1+ ρ
(1 − ε + rCD (G′[Y [t ]] − M [m[1 + t ]]))χ [1 + t ])
(4)
∂ CD Λ = 0 ⇔ μ[t ] = λ[t ]χ [t ] =
c[t ]U ′[c[t ]]
C[t ]
1
∂ m Λ = 0 ⇔ χ [t ] = −
M ′[m[t ]]
∂ s Λ = 0 ⇔ λ[t ] =
(5)
(6)
(7)
The first two equations are simply the state equations for K [t ] and CD[t ] . With
rK = (1 − Z [CD]) FK [ K , L] the marginal product of capital, the next equation tells us that in an
optimal program the current value of capital must be equal to its marginal benefit, which is the
discounted value of its net marginal product factoring in mitigation costs and net emissions
resulting from a larger capital stock. Since the time path of λ[t ] reports the change in the shadow
λ[t + 1]
price of capital, its change j[t ] =
− 1 yields the time path of the real interest rate, j[t ] .
λ[t ]
− Z ′[CD]Y
the marginal product of CD , the same has to hold in the fourth
With rCD =
Z [CD]
equation for CD : The price of CO 2 must be equal to the discounted value of its net marginal
product, again, factoring in mitigation positively and net emissions resulting from higher output
negatively. The last two equations are the Euler equations and establish optimality with regard to
the choice variables, s[t ] and m[t ] . They tell us, first, that marginal utility of consumption per
capita in period t has to be equal to cost of capital (which is measured in per capita utils per unit
of capital). Through equation (4) the marginal utility of consumption per capita is equal to the
per capita marginal benefits of accumulating more over the remaining time horizon; second, that
marginal cost of mitigating has to be equal to the marginal future benefit of doing so. The capital
(usually dollar) price of carbon emissions is fixed by the cost of the marginal emission reduction.
This thought will be taken up later. Given the Euler equations and the co-state equations, one can
derive Ramsey-Keynes rule equivalents of the system.
3.2 The business-as-usual case (BAU)
We model the business-as-usual case as an equilibrium of the economy in which global
warming is a public bad due to a negative externality. A state variable is an externality when it
has a real impact on the objective function or constraints, but no institutions exist to enforce the
social price on individual agent decisions involving it. Each agent assumes that her decisions will
not affect the path of the externality, but when all agents make the same decisions the path of the
externality changes. On a perfect-foresight equilibrium path with an uncorrected externality,
each agent is assumed to correctly forecast the path of the externality, but ignores the effect of
her decisions on the path of the externality. Thus on the equilibrium path with CD as an
uncorrected externality the typical agent solves the above maximization problem expecting a
certain time path of for the external CD[t ] . The correct forecasting assumption amounts to the
side condition that expected CD e [t ] = CD[t ] , where CD[t ] is the path of the externality
corresponding to the representative agent's chosen decisions. The difference between the
equilibrium path with an uncorrected externality and the optimal path is the fact that the typical
agent does not adjust her controls to take account of their effect on the externality. She is not
aware of the true social cost of emitting. Social institutions fail to provide the correct price signal
for the externality to steer the economy. As a result, the socially competitive equilibrium and the
optimum diverge. Also, the social bid price and social ask prices for the externality are not equal.
It is possible to express the second-best equilibrium path with an uncorrected externality
through the Lagrangian first-order conditions. The first-order conditions with respect to the
shadow prices return the real laws of motion of the system, which must be obeyed. The
representative agent, however, ignores the effect of her decisions on the external state variables,
which corresponds to setting the shadow price on these variables equal to zero in the first-order
conditions with respect to the non-external state and choice variables. The first-order condition
with respect to the external state variable then plays no active role in the solution, but does keep
track of the real social value of the externality in terms of its shadow-price μ . The BAU path
solves these modified first-order conditions:
∂ λ Λ BAU = 0 ⇔ K [t ] = (1 − δ ) K [t − 1] + s[t ]Y [t − 1]
∂ μ Λ BAU = 0 ⇔ CD[t ] = (1 − ε )CD[t − 1] + G[Y [t − 1]]
(8)
(9)
λ[t + 1]
(1 − δ + rK )
1+ ρ
∂ CD Λ BAU |μ [t ]= 0 = 0 ⇔ μ[t ] = λ[t ]χ [t ] = 0
(11)
∂ m Λ BAU |μ [ t ]= 0 = 0 ⇔ m[t ] = 0
(12)
∂ K Λ BAU |μ [ t ]= 0 = 0 ⇔ λ[t ] =
∂ s Λ BAU |μ [ t ]= 0 = 0 ⇔ λ[t ] =
c[t ]U ′[c[t ]]
C[t ]
(10)
(13)
It is important to notice the subtle differences between these equations and the fully
optimal path equations. The state equations (8) and (9) are the same on the optimal and BAU
paths. But on the BAU path the controls are optimized without taking account of the externality,
since μ[t ] = 0 in the first-order condition. As a result m[t ] = 0 . s[t ] remains unaltered due to the
specific form of the model ( s[t ] is the total share of income saved). Likewise, equation (10)
determines the shadow price of the non-external state variables with the shadow price of the
externality, μ[t ] = 0 . In the absence of the externality, the Euler equation for capital and the
optimal savings decision reduce to the usual Ramsey-Keynes rule.
There are two types of misallocation on the BAU path. First, because there is no market
price for carbon emissions, the typical agent allocates too little (zero) resources to mitigation.
Second, the typical agent invests too much in conventional capital because she ignores the
impact of increasing output on increasing climate damage.
In our calculations we keep track of how m[t ] and μ[t ] would evolve according to their
first-best first-order conditions given the other, second-best variables.
∂ m Λ = 0 ⇔ χ [t ] = M ′[m[t ]]
λ[t + 1]
(rCD (1 − m[t + 1]) +
1+ ρ
(1 − ε + rCD (G′[Y [t ]] − M [m[1 + t ]]))χ [1 + t ])
∂ CD Λ = 0 ⇔ μ[t ] = λ[t ]χ [t ] =
3.3 The constrained optimal case (COPT)
Researchers of the economic consequences of global warming (most prominently and
recently Nordhaus and Boyer (2000), Nordhaus (2008), and Stern (2007)) analyze the optimal
path in this type of model under the constraint that no mitigation is undertaken. Because this type
of path implicitly partially internalizes the externality, it seems to us that it does not represent
business-as-usual, and in the remainder of this paper we call this type of path constrained
optimal.
How exactly the constrained optimal path can be derived as an equilibrium within the
representative-agent perfect foresight methodology is somewhat mysterious. Fully rational
agents with perfect foresight acting with complete markets adopt the first best solution presented
above as the optimal path. When markets are incomplete and there is no price signal for the
marginal social value of the externality, the equilibrium is the BAU path described in the last
section. The constrained optimal (COPT) path, on the other hand, represents an inconsistent
mixture of assumptions about the representative agent's information. On the one hand, the
representative agent on this type of path correctly estimates the marginal social cost of emissions
in making her consumption, investment, and production decisions. On the other hand, she seems
to ignore the availability of mitigation technologies, despite this understanding of the marginal
social cost of emissions. This divergence results in a difference between the marginal social
value and marginal social cost of mitigation. The agents in this mixed scenario perceive the
marginal social cost of emitting as zero, the only price that justifies no mitigation. At the same
time, however, the agent is confronted with the true carbon price in her decision on how much
output to consume and how much to re-invest for capital formation. While this inconsistency
within the perfect foresight framework is corrected for in the business-as-usual case above, we
also solve for the constrained optimal case given its importance in the economic literature on
global warming.
The first-order conditions of the constrained optimal case are the special case of the fully
optimal with m = 0 .
∂ λ Λ COPT = 0 ⇔ K [t ] = (1 − δ ) K [t − 1] + s[t ]Y [t − 1]
∂ μ Λ COPT = 0 ⇔ CD[t ] = (1 − ε )CD[t − 1] + G[Y [t − 1]]
λ[t + 1]
(1 − δ + rK (1 + G′[Y [t ]]χ [t + 1]))
1+ ρ
λ[t + 1]
∂ CD Λ COPT = 0 ⇔ μ[t ] = λ[t ]χ [t ] =
1+ ρ
′
(rCD + (1 − ε + rCDG [Y [t ]])χ [1 + t ])
∂ K Λ COPT = 0 ⇔ λ[t ] =
∂ s Λ COPT = 0 ⇔ λ[t ] =
c[t ]U ′[c[t ]]
c[t ]
(14)
(15)
(16)
(17)
(18)
Notable changes include the altered state equation for CD[t ] in which carbon dioxide
concentration can only be altered through emissions and dissipation and the two modified costate equations. The change in the marginal benefit of capital from the optimal to the constrained
optimal path depends on the functional form of M [m[t ]] . For any meaningful mitigation
function, the marginal benefit of capital will be lower in the constrained optimal case. As a
result, less capital will be accumulated. This is intuitive as a reduction in K is the only means by
which CD[t ] increases can be counteracted with mitigation constrained to zero. Note that while
agents are deprived of the choice to mitigate, they still fully respond to changes in the price of
carbon in their accumulation decisions. The same logic applies to the price of CD[t ] . Lower
marginal benefit will lead to higher CO 2 concentration, as CD is a bad.
In our calculations below we retain the (for this program) superfluous optimality
condition for m[t ] in order to see what mitigation effort would be under an optimal scenario (at
that point in time of the program).
∂ m Λ = 0 ⇔ χ [t ] = −
1
M ′[m[t ]]
3.4 Basic Logic
The basic logic and qualitative features of the three cases can be seen even without
specifying functional forms, calibrating the model to reflect current economic values, and
solving it over long time horizons.
On the OPT equilibrium path agents will invest enough resources in mitigation to
compensate emissions to the point where marginal cost of doing so is equal to the benefit in
output due to less environmental damages. This level is defined by the damage function which is
key to the quantitative outcomes of the model. The carbon price will be defined by mitigation
efforts. It will be equal to the cost of marginal mitigation efforts.
On the BAU equilibrium path agents are not only deprived of the mitigation instrument,
but also see themselves incapable of affecting the stock of CO 2 . Their decisions are taken solely
with respect to maximizing their intertemporal utility and consumption. The carbon dynamics
drive the system. Emissions from rapid capital accumulation will drive up carbon dioxide levels
to the point where environmental damage chokes off further accumulation due to the falling
profit rate.
On the COPT equilibrium path agents are deprived of the mitigation instrument. In order
to move the economy to a steady state equilibrium, the savings decisions and the capital stock
will have to do all of the adjustment. It is important to note that in the COPT case (as well as in
the BAU case), the climate dynamics dominate the outcome. A steady state can only be achieved
when emissions equal to the dissipation of existing carbon stock in the atmosphere. This sets the
level of admissible capital stock and the appropriate savings rate.
Given the logic of the three model cases, it is trivial, but nonetheless important, to note
that the overall utility will be the greatest on the OPT path, followed by the COPT and BAU
paths. Growth in capital stock will be highest on the OPT, followed by the BAU. Investment will
be lower in the COPT than in the BAU scenario as agents are aware of the deleterious effects of
their accumulation decisions and, hence, more cautious. Climate catastrophes -- meaning high
equilibrium levels of CO 2 -- are certain in the BAU and very likely in the COPT case. Over the
range of all realistic mitigation functions, CO 2 levels will stabilize at low or moderate levels in
the OPT case.
These characteristics are given by the model structure and are, thus, independent of the
functional forms and parameter values. We hope that in this light the secondary relevance of
much of the current debate on discounting factors will be apparent.
Fast convergence to the steady state implies that the steady state results will drive much
of the model's behavior and its transition dynamics. As can be seen below, the optimal path
reaches its steady state within 10 decades. Nordhaus (2008) and Stern (2007) do not devote much
attention to the steady states of their models and the implications of steady state values on the
transition dynamics.
4 Functional Forms
There are several choices that need to be made about the forms the production, damage,
mitigation, and emissions functions take. The functional forms we use in the simulations
c[t ]1−η
reported here are as follows: Utility has its traditional iso-elastic manifestation U [c[t ]] =
,
1 −η
or U [c[t ]] = Log[c[t ]] when η = 1 . Potential output is a Cobb-Douglas production function,
F [ K [t ], L[t ]] = AK [t ]α L[t ]1−α . Carbon-related damages are measured on a scale between 0 and 1
with zero damage at the pre-industrial level of 280 ppmv and complete output loss at a
γ
1
⎞
⎛
⎜ ⎛ CD[t ] − 280 ⎞ γ ⎟
CDMax = 780ppmv , with the damage function Z [CD[t ]] = 1 − ⎜1 − ⎜
⎟ ⎟ . This
⎜ ⎝ CDMax − 280 ⎠ ⎟
⎠
⎝
functional form implies that even at current CO 2 levels of 380 ppmv a certain fraction of
potential output is lost due to environmental degradation. Emissions, G[Y [t ]] = βY [t ] , are linear
in output at a constant carbon intensity of production. The mitigation function,
1 − e −νm[ t ]
, where ζ is a scaling parameter and 0 < ν is a semi-elasticity reflecting
M [m[t ]] = ζ
ν
diminishing returns to m , which converts the unitless proportion of output devoted to mitigation,
m , into CO 2 reduction per $ spent on mitigation, that is ppmv/$ . In our specification of the
mitigation function, we diverge from the other studies in assuming positive mitigation costs even
at very low mitigation efforts ( ∂ m M [m[t ]] |m[t ]=0 = ζ ≠ 0 ). From the first-order condition for m[t ] ,
M ′[m[t ]] equals the carbon price χ [t ] in the first best solution.
Population growth follows Nordhaus (2008) and UN projections in assuming that world
population will rise from currently 6500 to 8600 million over the next 10 to 20 decades, and then
stabilize at this level. Labor productivity is assumed to start at a yearly growth rate of 2% and to
flatten out at 3 times its current value after 30 decades.
Given these functional forms, the adjoined Lagrangian (with L[t ] = B[t ]N [t ] ) is:
T
Λ[ K [t ], CD[t ], λ[t ], μ[t ], t ] = ∑
t =1
1
(1 + ρ ) ( t −1)
⎛ ⎛ (1 − s[t ]) Z [CD[t − 1]]F [ K [t − 1], L[t − 1]] ⎞ (1−η )
⎜⎜
⎟⎟
⎜ ⎜⎝
N [t ]
⎠
+ λ[t ]( K [t ] − (1 − δ ) K [t − 1]
⎜
1 −η
⎜
⎜
⎝
+ ( s[t ] − m[t ]) Z [CD[t − 1]]F [ K [t − 1], L[t − 1]])
⎛
⎛
1 − e −νm[t ] ⎞
⎟⎟
+ μ[t ]⎜⎜ CD[t ] − (1 − ε )CD[t − 1] + ⎜⎜ β − ζ
ν
⎝
⎠
⎝
Z [CD[t − 1]]F [ K [t − 1], L[t − 1]]))
5 Calibration
Given the assumptions of functional forms, the actual functions need to be calibrated to
match economic and physical realities. All parameters are geared towards a decadal time
interval. In the benchmark case Lagrangian, the discounting factor, ρ = 0.1 . With output
measured in units current (2000-2010) $ trillion, initial capital stock is assumed to be K 0 = 200 .
CO 2 is measured as parts per million volume (ppmv). Initial CO 2 concentration CD 0 = 380 .
Capital decays radioactively with δ = 0.7 .
Currently, 7 Gt carbon are burnt per year. This corresponds to an increase in CD of 3.37
ppmv. With a yearly world output of $ 60 trillion this implies a carbon dioxide emission intensity
3.37
ppmv
. As the actual increase in atmospheric carbon dioxide concentration
= 0.0561
β=
60
$ trillion
is only about 2 ppmv, dissipation is 1.37 ppmv. This yields a depreciation factor
1.37
ε=
= 0.0036 .
380
The marginal product of capital in the Cobb-Douglas production function α is set at
0.35 in line with standard economic research. Total factor productivity is calibrated to match
current world output of $ 60 trillion. The elasticity of substitution in the utility function η is set
at 2 , in our baseline simulations.
The damage function Z [CD[t ]] also takes an iso-elastic form. This allows us to combine
the apparent global warming optimism of economists towards the low damages of global
warming at low carbon dioxide concentration with the serious warnings of climate scientists
about severe output loss at high carbon dioxide concentration (which is set to 780 ppmv in our
model). We set γ = 0.5 , which is at the higher end of potential damages (Barker, 2009). The
calibration points usually cited are the IPCC (2008) predictions of an increase to a doubling of
pre-industrial concentrations (about 280 ppmv) leading to a temperature rise of 3 o C and an
increase of 4 o C leading to a potential output loss of 1 − 5% of current output. Nordhaus (2008)
assumes that current damages to the world economy are 0.15% of output. This corresponds to
γ = 0.3 . Note that the results for γ = 0.3 are reported in the sensitivity analysis section below.
We deviate from Nordhaus as his damage function is lacking any reasonable upper limit on
temperature and ultimately CO 2 concentration. Rezai (2008) shows that our parametric form of
a damage function lies consistently above Nordhaus' for γ = 0.5 and below it with γ = 0.3 up to
500 ppmv (which is the relevant range for the optimal program). Figure 1 plots the damage
function for γ = 0.5 and a band of 0.7 ≥ γ ≥ 0.3 , around which a sensitivity analysis is carried
out below. It also includes the assumptions on environmental damages from Nordhaus (2008) as
a gray area up to CD = 580 . Note that while our assumptions on γ regarding damages can be
regarded as high, so are the assumptions on mitigation costs with the carbon price at $ 160 (per t
of C).
Figure 1: Damage Function with
γ = 0.5
The parameter ζ represents the marginal reduction in CO 2 concentration (ppmv) per
$ T from spending a small amount on mitigation when mitigation is zero. If it costs $ 160 to
remove one tonne of C at present (carbon markets suggest level between $ 75 and $ 125), when
effectively m = 0 , then to reduce CO 2 concentration by 1 ppmv through removing 2.07 Gt C
ppmv
$ T . Using
from emissions would cost (2.07)(0.160$trillion) = 0.331 $ T , so we set ζ = 3
this specification the carbon price is directly linked to and anchored by marginal mitigation
efforts. Note that the assumption of a lower current carbon price increases ζ and makes the
mitigation function more effective.
Figure 2: Mitigation Function for
ζ =3
and ν = 6
The table below provides an overview over the parameter assumption.
Function
Parameter
Value
ρ
0.1
Λ [.]
200
K0
CD 0
δ
U[.]
F[.]
Z[.]
G[.]
M[.]
ε
η
α
A
γ
CDMax
β
ζ
ν
380
0.7
0.0036
2
0.35
28.4
0.5
780
0.056
3
6
Table 1: Parametric Overview
6 Computational Implementation
The above systems of equations can be reduced to 4 laws of motion, 2 for the state
variables and 2 for the co-state variables by substituting the optimal expressions for the controls
into the other equations and forming thus the maximized Lagrangian. These 4 difference
equations, of which some form a subsystem in the COPT and BAU case, have to hold for
t = 1,..., T . In addition, initial conditions on the two state variables and terminal transversality
conditions on the co-state variables have to hold. This yields 4(T + 1) conditions to determine
4(T + 1) variables.
In order to solve this set of equations, we make use of the program Mathematica and its
root finding command. In this process the specification of initial search parameters is crucial.
Generally, Mathematica proves to be quite agile in finding the equilibrium path even if state and
co-state variables are persistently shifting on a steady growth path. Specifying variables in
logarithmic forms some times facilitates the search routine in this case.
7 Growth Paths
The above first-order conditions are sufficient as well as necessary for a global maximum
as the maximized Hamiltonian is concave in K and CD for any given λ and μ (that is, the
objective function is quasi-concave and the constraints convex in K and CD ). The programs
below are set up as finite-horizon problems with the additional requirement of the transversality
conditions λ[T ]K [T ] = 0 and μ[T ]CD[T ] = 0 with a fixed T at 60 decades. These terminal
conditions guarantee that the calculated OPT path is a valid optimum of the primal problem over
the finite time horizon. We choose the time horizon sufficiently long such that the paths to
approximate the steady-state in the middle of the time horizon. This characteristic is known as
the turnpike property which assures that finite horizon programs mimic their infinite horizon
twins for sufficiently long horizons (Samuelson, 1965). The paths below effectively reach their
steady state within 30 decades. Solving over 60 periods becomes sufficiently close to the infinite
horizon problem. Note that it would be possible to change the transversality end-point conditions
to require, for example, some minimal capital and maximal CO 2 stocks at the end of the
program, but it is not very easy to see how to choose those levels, or, in fact, to use this method
except in the finite time horizon case. (It is not really correct to force the path to converge to the
steady state at any finite time, for example.)
Figure 3 is a comparison of the optimal, constrained optimal and business-as-usual paths
for these parameters, in terms of world per capita consumption, the damage from global
warming, the implied price of carbon, and the CO 2 concentration. The finite-horizon program
approaches its steady state very quickly, as to be expected from the turnpike theorem.
Figure 3: The optimal equilibrium path, OPT, is plotted in green, the BAU equilibrium in red, and the constrained
equilibrium, COPT, in blue. The carbon price on the optimal path is about $ 200/t, and the damage on the optimal
path is less than 1 % of potential output. The carbon price is the social marginal value of foregoing the emissions
from 1t of carbon. For OPT and COPT the carbon price is effective in economic decisions, but not in BAU (and is
plotted as a dashed line). For OPT the mitigation percentage is the proportion of world product devoted to
mitigation. For COPT and BAU the mitigation percentage is the investment called for by the imputed carbon price
(and is plotted as a dashed line), while the actual mitigation is zero.
On the unconstrained optimal path capital accumulation combined with small mitigation
efforts of around 1− 2% of GDP enables sustainably rising output and consumption levels.
Mitigation efforts are front-loaded, meaning that most of the mitigation is done in the first few
decades, such that only current CO 2 emissions have to be mitigated in later periods. The carbon
price stabilizes around $180 /t which is close to zero mitigation carbon cost of $ 160. In fact,
atmospheric carbon concentration decreases almost to its pre-industrial level.
On the BAU path agents lack the correct price signals to correct the negative externality.
This leads to inefficiencies in several respects. Capital rises at a rate similar to the optimal case
during the first 100 years although no mitigation can be carried out; the implication of this overaccumulation is rapidly rising CO 2 concentration and environmental damage. Since the capital
accumulation equation for λ[t ]BAU is independent of any carbon related costs (or their price
signals), accumulation continues until damages are so high that further accumulation cannot
occur due to the declining productivity of capital and labor inputs. Output and consumption are
bound to decline due to ever higher carbon concentration and damages until the marginal product
of capital has fallen sufficiently to approach a stable equilibrium. Equilibrium output and
consumption per capita are almost 20% below current levels and less than 25% of the OPT
levels despite significant technical progress and population growth. The red dashed lines in the
carbon price and mitigation graphs depict the implied carbon price and the mitigation called for
by this price. Implicit carbon price and mitigation efforts on the BAU are fifty times higher than
their optimal counterparts. The inefficiency of the BAU can also be seen in the equilibrium
saving rates. While higher capital stock implies higher saving to compensate the lower marginal
product, the equilibrium BAU saving rate with its lower capital stock is higher in our simulations
due to the (unnecessary) high carbon concentration.
The constrained optimal path does slightly better than the BAU path. Although the agents
in the COPT also are confined to zero mitigation, they conceive the correct price signals and run
up GHG in the atmosphere much more cautiously than in the BAU scenario. In fact, overall
capital stock is decreasing on this path as current levels are suboptimally high, implying too
many carbon emissions. As carbon concentration increases, so must the marginal product of
capital which can only be achieved with a lower capital stock. Equilibrium GHG concentration
entails a high carbon price which, again, calls for high mitigation efforts. Equilibrium carbon
concentration and price are lower than on the BAU path. The carbon price is about twenty times
the optimal level.
While the quantitative results of our simulations are dependent on the specific parameter
assumptions, it is important to point out that the qualitative results are independent of these
assumptions. Especially the finding that moving from the inefficient BAU path to the efficient
OPT path through mitigation constitutes a Pareto improvement should be noted. This result
implies that there is no cost to mitigation, but there are significant gains from doing so (in our
simulations up to 400% of GDP). Higher or lower discounting rates will not alter this. In light of
the magnitude of the avail, we argue that questions of uncertainty and intergenerational equity
which are discussed further below can be considered of second-order importance.
8 Intergenerational Equity
An important aspect of the current climate change debate in the economic literature
centers on the problem of intergenerational equity. This focus on generational equity arises
primarily from a failure to appreciate that the business-as-usual path with an uncorrected
externality is inefficient. In particular, it is a consequence of the mistaken use of the COPT path
rather than the real BAU path as the benchmark with which the OPT path is compared. The most
striking difference between the COPT and OPT paths is the generational distribution of
consumption. As we have explained above, the COPT path is not a theoretically relevant
benchmark, because it represents an inconsistent mixture of partial internalization of the global
warming externality and a failure to divert resources from conventional investment to mitigation.
The use of the COPT path as the benchmark comparison leads to the misleading impression that
the problem of correcting the global warming externality is primarily an issue of
intergenerational equity, how much to sacrifice the consumption of current generations to protect
the environment for future generations. In an optimal growth framework the resolution of this
trade-off depends on the discount factor, ρ , and the degree of social preference for consumption
smoothing expressed in the elasticity of felicity with respect to consumption, η . But with the
correct BAU benchmark the correction of the global warming externality can provide an
intergenerational Pareto-improvement, raising the per capita consumption and felicities of every
generation. The parameters ρ and η influence the distribution of this intergenerational gain,
which is a second-order consideration. In fact, with our benchmark values of ρ = .1/decade and
η = 2 , the OPT path exhibits slightly lower per capita consumption than BAU in the first decade.
One way to understand this fact is that the representative agent in the BAU equilibrium values
current consumption too highly relative to investment in mitigation; when she grasps the full
social marginal value of mitigation she prefers to reduce her consumption slightly in the first
decade because of the high rate of return of this investment to the consumption of future
generations. If the representative agent put more weight on intergenerational smoothing of
consumption, for example, if η = 3 , the corresponding OPT path would dominate the BAU path
for η = 2 , thus demonstrating the inefficiency of the BAU equilibrium.
In Figure 4 below we plot the initial decades of four paths to underline these points. The
COPT, and BAU paths relative to the OPT path all of which are the same as those plotted in
Figure 3 and calculated with the benchmark parameters, in particular with η = 2 . The OPT' path
has the same parameters for the technical side of the model, but sets η = 3 , to emphasize the
inefficiency of the BAU path. The figure shows that the use of the COPT equilibrium as the
benchmark distorts the perception of the economic issues involved in global warming policy by
incorrectly suggesting that correction of the global warming externality will depress per capita
world consumption significantly for several decades.
Given the inefficiency of over-accumulation of GHG stock in the atmosphere as a result
of capital stock accumulation, the world economy is not operating at the intertemporal
production possibility frontier (PPF). Future generations would appreciate lower stocks of CO 2
which implies that current generations should accumulate less conventional capital and consume
more (of it) today. There is no intergenerational trade-off as is posited in most of the global
warming related economic literature. The mutual gains are illustrated by moving the economy
from an inefficient point inside the PPF (the BAU path) to its boundary (the OPT' path with
η = 3 ).
Figure 4: Comparisons of the consumption paths for the business-as-usual, constrained optimal, and the altered
optimal (with η = 3 ) cases. All normalized to the optimal consumption stream.
Consumption in the COPT case lies above consumption in the optimal case in the first
three decades as agents who are aware of the deleterious effects of global warming wisely
choose to accumulate less and consume more. This positive difference in the first few decades
forms the basis for the intergenerational equity discussion since the choice of the appropriate
program now depends on its parameterization (most importantly the discount factor and the
elasticity of felicity with respect to per capita consumption). Nordhaus (2008) and Stern et al.
(2007) place much emphasis on the dominance of baseline consumption over optimal
consumption in the first few decades. This implies that current generations would attain lower
utility levels if they started investing into mitigation. The consumption paths of the optimal and
business-as-usual scenarios virtually move together in the first few decades with the BAU
dominating the OPT path for the first two decades. The OPT' path, however, shows that this
effect is due to the representative agent preferring to transfer some of the gain from globalwarming mitigation to future generations given the very high rate of return to mitigation
investment given the initial conditions.
9 Parameter Sensitivity
In order to gauge the effect of parametric changes on the quantitative results of our
simulations, this section explores sensitivity analysis for some model parameters. Generally,
consumption smoothing dominates most of the adjustment in parametric changes. Due to spatial
limitations, only the selected parameters are reported here. A complete set can be obtained from
the authors. Table 2 presents an overview over the effects of parametric changes (rows) on
selected variables (columns) relative to their values with standard parameters (which are reported
in the bottom row). The sensitivity table reports lower and upper bounds for each parameter.
For example, the first table in table 2 captures the effects on the OPT path. The first two
rows report changes in the discount rate of which the standard value is 0.1 . The first row has
ρ = 0.05 and the second ρ = 0.15 . The first column reports the changes on consumption per
capita. A lower time preference increases consumption by 0.5% at the median and the 20 %
quantile compared to the consumption stream under standard parameter assumptions.
Consumption rises by 0.6% above the corresponding standard consumption level at the 80 %
quantile. An increase in the discount rate to ρ = 0.15 decreases consumption by 0.6% at the
20 % quantile and by 0.7% at the median and 80 % quantile. The next column reports effects of
the changes on the carbon price. The change in pure time preference has virtually no effect on
the carbon price. The third column reports the effects of parametric changes on mitigation
efforts, the fourth on atmospheric carbon concentration, and so on. The third and fourth row the
changes in the severity of damage, γ , the fifth and the sixth changes in the upper bound of
atmospheric changes in CD , and so on.
The table for the BAU and COPT paths do not report parameter changes in ν and ζ ,
since these parameters concern the mitigation function and there is no mitigation available in
these scenarios.
Varying the discount factor has very different effects in the three scenarios. In the
optimal case higher time preference yields higher consumption today and higher CO2
concentration and damage in the future. This is the standard result which most of the
intergenerational literature rests on. In the BAU the higher consumption induced by higher time
preference leads to less capital accumulation and less carbon emissions which implies less
carbon damage later on. This lowers the extent of the climate crisis. A lower valuation of the
future, thus, leads to (marginally) higher consumption today and tomorrow as the externality is
mitigated. In the constrained optimal case the increase in ρ leads to the standard intertemporal
reallocation. A higher consumption level is sustained by higher accumulation with higher carbon
concentrations early on. The shift also implies that the marginal rates shift. In the future a higher
damage level has to be compensated with higher maintenance investments. Consumption is,
consequently, lowered.
A reduction in γ to 0.3 leads to lower environmental damages. In the optimal case
capital, output, and consumption can rise, while mitigation efforts fall. For marginal rates to
equalize, CO 2 concentration and damages rise. In the constrained optimal case lower damages,
again, lead to higher output and consumption and higher CO 2 concentrations. Damages and
capital stock, however, fall. A milder damage function allows higher GHG concentrations in the
atmosphere although these are not run up to the same damage level as before. Less damage
requires less capital stock for the same level of consumption and less maintenance investment.
The resources freed increase consumption even further. In the BAU case a milder damage
function implies a milder climate crisis. More capital and CD can be accumulated. In the BAU
capital accumulation is only slowed down by falling profitability due to environmental damages.
A softer damage function allows a higher level of consumption to be obtained before and after a
climate catastrophe. The climate crisis itself is more pronounced as can be seen in the steep fall
of output and the reversal of the interest rate to almost − 7% . World output and consumption
increase while CO 2 is run up, but fall more drastically thereafter.
Table 2: Effects of Parametric Changes on Time Paths of Selected Variables Relative to their Standard Value
In the optimal case agents are able to fully internalize the externality. As a result the
effects of increases in the maximal permissible CO 2 level are negligible. In the BAU the
increase of CDMax increases output and consumption considerably. Generally the transition to
the steady state occurs much smoother which is due to the relative decrease of carbon emissions
to carbon dissipation at higher CO 2 levels. This allows a higher capital stock and higher
economic activity. The same rationale holds true for the constrained optimal case. Capital stock,
output and consumption increase together with CD , while damages decrease.
The mitigation function is only relevant in the optimal case and consists of two
parameters. ζ measures the cost of mitigation as it ties marginal mitigation to the carbon price.
ν measures the effectiveness of mitigation for different levels of mitigation investment, m . An
increase in the cost structure of mitigation leads to a higher carbon price and higher mitigation
costs, with slight increases in CD and environmental damages to offset some of the cost
increase. A decrease in the effectiveness of mitigation leads, again, to increased mitigation costs
and a higher carbon price. This movement is counteracted by a slight increase in CO 2 and
environmental damage. Note that front-loading of mitigation efforts is not a robust result and can
change depending on parametric assumptions regarding the mitigation and damage functions.
Convergence to full mitigation levels remains robust at 6 decades.
10 Conclusions
Human emissions of carbon dioxide into the atmosphere are a negative externality;
individuals do not take into account the impact of their emissions on atmospheric CO 2 , but
atmospheric concentration of CO 2 affects their (or their descendants') collective well-being
through the deleterious effects of global warming. Such a negative externality leads to a market
failure and an inefficient allocation of resources. In the case of global warming the inefficiency
takes the form of over-accumulation of capital and under-investment in mitigation. The
correction of this market failure through the implementation of institutions which enforce cost
transparency represents a Pareto improvement: the current generation invests less, spending the
retrenchment on consumption and mitigation such that future generations enjoy higher output,
higher consumption combined with lower GHG concentration.
Our simulations show that the gains from such a movement to the intertemporal
production possibility frontier are large. While this reallocation will imply significant changes of
our life-style and the technology used, the mitigation costs to return to pre-industrial carbon
dioxide levels are, however, relatively small with their peak at around 2% of world product.
The question of how such cost transparency can be achieved and which mechanisms to
use are important, but cannot be answered in our framework. This is also true for the Pareto
improvement question of who needs to compensate whom. Given our representative agent
assumption, such a realistic question is clearly unanswerable. These problems will need to be
solved by the world political system.
If global warming is a real economic problem, there is no economic cost to correct it. In
principle, the costs of reducing emissions in the current generation can be shifted to the future
generations who will benefit from a cooler planet by reducing conventional investment.
11 Acknowledgments
Support from the Schwartz Center for Economic Policy Analysis is gratefully
acknowledged.
12 References
Barker, T.: The economics of dangerous climate change. Climatic Change (forthcoming)
Foley, D.K.: The economic fundamentals of global warming. In: Harris, J.M. and
Goodwin, N.R., (eds.) Twenty-First Century Macroeconomics: Responding to the Climate
Challenge, ch. 5. Edward Elgar Publishing, Cheltenham and Northampton (2008)
IPCC: Climate Change 2007, the Fourth IPCC Assessment
http://www.ipcc.ch/ipccreports/ar4-syr.htm (2007). Accessed 10 August 2008
Report.
Nordhaus, W.D.: A Question of Balance: Weighing the Options on Global Warming
Policies. Yale University Press, New Haven (2008)
Nordhaus, W.D., Boyer, J.: Warming the World. MIT Press, Cambridge (2000)
Rezai, A.: Recast DICE: the economics of global warming. mimeo (2008)
Samuelson, P.A.: A catenary turnpike theorem involving consumption and the golden
rule. A E R 55, 486-496 (1965)
Stern, N.: The Economics of Climate Change: The Stern Review. Cambridge University
Press, Cambridge (2007)